Yeah something is not right. I logged in and had a look.
Even the native WP customizer isn’t working.
I think there may be an issue with your install of WordPress.
I would re-install WP from scratch, and then re-install the theme.
I tested with the default theme and it has the same issue.
I have never encountered this issue before, so that is where I would start.
If that doesn’t solve it let me know so I can debug further.